The training part of the neural network involves a large number of derivation operations due to the back-propagation phase and includes nonlinear functions, such as Softmax. Compared with the inference part, the training part is more difficult and has less research[1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}. The earliest research to realize the neural network training is SecureML, which improves the generation of multiplication triplets, uses Secret Sharing for linear computation, Garbled Circuit for nonlinear computation, such as comparison, and uses ReLU to construct the approximate functions of Sigmoid and Softmax. Then, the training of neural network is realized for the first time, but the time and accuracy of the model still have much room for improvement. QUOTIENT[4]} tailors the weight to \(\left\lbrace  { - 1,0,1} \right\rbrace \) , uses MPC to construct the quantization and normalization in the back propagation of the neural network, and designs the optimization algorithm in the fixed-point domain based on the adaptive gradient optimization. ABY\(^3\)  improves the conversion among arithmetic sharing, Boolean sharing, and Yao sharing and realizes the training of neural networks on 3PC for the first time. Based on the basic framework of SecureML and adding a third-party assistant, SecureNN proposes a new comparison protocol based on Secret Sharing and builds secret sharing ReLU and Maxpool functions based on this. FALCON further constructs a 3PC neural network training method with equal status of the three parties on the basis of SecureNN and ABY\(^3\)  and improves the comparison protocol and linear computation for 3PC, which has greatly improved time efficiency and framework security. However, the improved comparison protocol still requires the number of multiplications proportional to the data bits, so there is still room for improvement. Moreover, the three recent papers are all research on 3PC. To the best of our knowledge, only a few studies on the neural network training of 2PC have been performed in the past two years.
